The directional zero-one law for random walks in iid random environments in dimension at least three
Consider nearest-neighbour random walk on in an environment of iid transition rows, and for a direction let . Kalikow (1981) proved for the annealed law, and the question whether itself must be 0 or 1 goes back to him. Zerner and Merkl (2001) proved it in dimension two for iid strictly elliptic environments; Bramson, Zeitouni and Zerner (2006) showed it fails for some stationary mixing environments in and posed the iid case as Open Problem 3. Is the annealed escape probability always 0 or 1 for iid elliptic environments in every dimension ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Random walks in random environment
- Posed by
- Steven A. Kalikow; the iid higher-dimensional case stated as Open Problem 3 by Maury Bramson, Ofer Zeitouni and Martin Zerner
- Year posed
- 1981
- Years open
- 45y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 36 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for , iid nearest-neighbour rows that are almost surely strictly positive (no uniform lower bound, no moment condition) and any fixed real , . The October companion extends the law to stationary ergodic finite-range-dependent environments under uniform ellipticity, and the family's ballisticity paper gives, for uniformly elliptic iid environments in , the hemisphere description of the transient directions. Not shown: ballisticity or a velocity under strict ellipticity alone, jumps beyond nearest neighbours, or general mixing environments (where the law fails by Bramson-Zeitouni-Zerner).
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts (September 23 and October 5, 2026).
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript were read against the question as cited. The proof (regeneration in a real direction, finite spatial radii under coexistence, and an entropy bound on contacts between regeneration arrangements) was not refereed. The challenge lean/ComparatorChallenges/DirectionalWalk.json (theorem OAI.DirectionalZeroOne.directional_zero_one, solution module OAI.Probability.DirectionalWalk.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; it is found through lean/docs/220.md. Its statement was read here: for , any probability law on nearest-neighbour transition rows whose entries are almost surely all positive, the iid product environment, and any real , the annealed probability from the origin that is 0 or 1. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.